be a \(J\) -holomorphic map, such that \([u] = 0\)  in \(H_2(X,X \setminus D)\) .
Then \(u\)  is contained in \(\lbrace h \le c\rbrace \) .


Follows from the integrated maximum principle of [1]}.
Namely, find a sequence \((\epsilon _k)\)  of numbers in \((0,\delta )\) , decreasing to zero, such that for each \(k\) , \(\Sigma \)  intersects \(\lbrace h=c+\epsilon _k\rbrace \)  transversely.
For any \(k\) , let \(u_k:(\Sigma _k,\partial \Sigma _k) \rightarrow (\lbrace h\ge c+\epsilon _k\rbrace ,\lbrace h=c+\epsilon _k\rbrace )\)  be the part of \(u\)  mapping to \(\lbrace h\ge c+\epsilon _k\rbrace \) .
We have \([u_k] = [u] = 0\)  in \(H_2(X,X \setminus D)\) , by assumption; hence, \([\omega ;\alpha ](u_k) = 0\) .
It follows by definition that
\( \omega (u_k) = \alpha (\partial u_k).\) 
